Welcome to the forum!
You can do the following steps to help you to solve your concern:
1. Switch off your phone.
2. Take out the SIM card and battery for a second.
3. Switch off your phone.
4. Open the Internet sharing if it will turn on.
5. If it's still the same, you can try inserting another SIM card to isolate the issue.
If in case that the issue still persists, try to reset your Lumia 620 by pressing and holding both the volume down and the power key for 10 seconds. If you've done that before, you can also try the other way of resetting the phone. This procedure will delete all of your files. So, back up is needed.
Here's how you can back up your files: http://www.windowsphone.com/en-US/How-to/wp8/basic
To reset your phone, go to Settings > About > Reset your phone.
If the above steps didn't work, you may update the phone's software. See this link for more information:
Hope this helps.